In the previous lesson, you learned how to read the yield curve. Today, it leaves the chart and moves into the calculator: we are going to price our first bond, the Tesouro Prefixado, known in the market by its technical name, LTN (Letra do Tesouro Nacional).By the end of this lesson, you will be able to answer three questions that every fixed-income investor should understand — and that most do not: how much is a fixed-rate bond worth today? Why does its price fall when interest rates rise? And why does the Tesouro Direto app sometimes show your position in the red, even for a "risk-free" bond?1. The simplest bond in the worldAn LTN is what the market calls a zero-coupon (or "bullet") bond: it pays nothing during its life and makes a single payment at maturity, the nominal value of R$ 1,000.00. Always R$ 1,000, for any LTN, at any maturity.If the final value is fixed, where does the return come from? From the discount: you buy the bond today for less than R$ 1,000, and the difference between that price and the R$ 1,000 redemption value is your return. The lower the purchase price, the higher the embedded yield — and vice versa.That is why the LTN is the perfect laboratory for learning how bonds are priced: a single cash flow, a single calculation, and the entire logic of fixed income laid bare.An LTN maturing in 2029 and another maturing in 2032 will be redeemed at maturity for what amount?Answer: R$ 1,000.00 each, regardless of maturity — the nominal value of an LTN is always R$ 1,000. What changes between maturities is today's purchase price: the longer the term, the greater the discount.2. The formula: discounting R$ 1,000 to present valuePricing an LTN means answering the question: how much is the right to receive R$ 1,000 at maturity worth today? It is simply Lesson 2: present value with compound interest, using the Brazilian convention of 252 business days.P = 1.000 / (1 + i)^(du/252)Where P is the purchase price (called PU, or unit price), i is the annual rate at which the bond is traded and du is the number of business days until maturity.2.1 Worked exampleAn LTN matures in exactly 2 years (504 business days) and is trading at 12% a year. How much does it cost?P = 1.000 / (1,12)^(504/252) = 1.000 / (1,12)² = 1.000 / 1,2544 = R$ 797,19In other words, if you pay R$ 797.19 today and receive R$ 1,000.00 two years from now, your return is exactly 12% a year, in exchange for leaving your money invested until then.On an HP-12C: 1000 [FV], 12 [i], 2 [n], [PV] → −797.19. In Excel: =1000/(1+12%)^(504/252) or =VP(12%;2;;1000).2.2 The reverse calculation: from price to yieldThe market also performs the calculation in reverse. If a 1-year LTN (252 business days) is trading at R$ 892.86, what yield is embedded in its price?i = (1.000 / 892,86)^(252/252) − 1 = 1,12 − 1 = 12% a yearPrice and yield are two ways of expressing the same thing. On trading desks, the yield is negotiated; the PU is the result.An LTN matures in 252 business days and is trading at 10% a year. Its unit price is approximately...Answer: R$ 909.09 — P = 1.000/(1,10)^(252/252) = 1.000/1,10 = 909,09. The answer "R$ 900.00" is the trap for anyone discounting using simple interest (1.000 − 10%): remember, the convention is compound interest!3. The seesaw: why price and yield move in opposite directionsLook at the formula again: the yield is in the denominator. The inevitable result:Yield rises → price falls. Yield falls → price rises.This inverse relationship is perhaps the most important concept in all of fixed income — and the least intuitive for beginners. Let us look at it in numbers, continuing with the example of the 2-year LTN bought at 12% (R$ 797.19):If, the next day, the market yield for that maturity rises to 14%: P = 1.000/(1,14)² = R$ 769,47. Your bond is now worth R$ 27.72 less (−3,5%).If, instead, the yield falls to 10%: P = 1.000/(1,10)² = R$ 826,45. Your bond has gained R$ 29.26 (+3,7%).The economic intuition is straightforward: if the market is now offering new bonds yielding 14%, no one will pay the old price for your 12% bond — for it to "yield 14%" to a new buyer, its price has to fall. The old bond adjusts to the new environment through its price.After an investor buys an LTN, market interest rates for that maturity rise. The bond's market price...Answer: Fell, because price and yield have an inverse relationship — the yield is in the denominator of the present value. Higher new interest rates devalue an old bond that pays a lower yield.4. Marking to market: the statement in the redYou can now understand the scene that frightens thousands of investors: someone buys Tesouro Prefixado, "the safest investment in Brazil," opens the app two months later and sees a negative return.What happened? Marking to market: Tesouro Direto (as well as every investment fund, by regulatory requirement) updates the value of your position every day based on the price the bond would fetch if sold today — calculated using the current yield curve, the one you learned to read in Lesson 3. If the curve shifted higher after you bought, the PU fell, and the statement shows a potential loss.Potential is the key word. This brings us to the two golden rules of fixed-rate bonds:Rule 1 — Hold it to maturity: you receive exactly the contracted rate. The R$ 1,000 at the end does not change; whoever bought at 12% and held the bond earned 12% a year, regardless of what happened to the curve along the way.Rule 2 — Sell before maturity: you receive the day's market price. It may be more (if rates have fallen) or less (if they have risen) than the contracted return. The entire "risk" of a fixed-rate bond lies here.An investor bought an LTN at 12% a year and held it until maturity. During that period, market rates fluctuated between 10% and 15%. What annual return did the investor actually earn?Answer: 12% a year, the rate contracted at the time of purchase — by holding the bond to maturity, the investor receives the R$ 1,000 nominal value, and the calculation based on the purchase price and redemption value produces exactly the purchase yield. The fluctuations in between only affect anyone who sells early.Exam (and real-life) trick: "a fixed-rate bond has a guaranteed return" is conditionally true — guaranteed AT MATURITY. Before then, the bond fluctuates every day with the yield curve. Fixed income does not mean a fixed price!5. Maturity matters: the leverage effect of time to maturityCompare two LTNs under the same interest-rate shock, with the rate jumping from 12% to 14%:1-year LTN: price falls from R$ 892.86 to R$ 877.19 → a 1,8% decline.5-year LTN: price falls from R$ 567.43 to R$ 519.37 → an 8,5% decline.The same shock causes nearly five times as much damage to the longer bond. It makes sense: with a 5-year bond, you lock in the old rate for much longer — the discount accumulates in the exponent. The longer the maturity, the more sensitive the price is to changes in yield.This intuition has a name, a surname and a formula: it is called duration, and it is the subject of Lesson 7. For now, remember the principle: a longer maturity amplifies both the gains and the losses from marking to market.Two LTNs, one maturing in 1 year and the other in 5 years, experience the same increase in market interest rates. Regarding the percentage change in price, it is correct to say that...Answer: The 5-year LTN falls proportionally more than the 1-year LTN — the longer maturity amplifies the effect of the rate on the compound discount. This is the seed of the duration concept, which we will formalize in Lesson 7.“At maturity, a fixed-rate bond keeps its promise. Along the way, it demands courage.”- Fixed-income trading-desk saying6. Putting it all togetherThe essential points of the lesson in five lines:The LTN (Tesouro Prefixado) is a zero-coupon bond that pays R$ 1,000 at maturity; the return comes from the discount.Its price is the present value of the nominal amount: P = 1.000/(1+i)^(du/252) — and the same formula can be used to derive the yield from the price.Price and yield have an inverse relationship: the yield sits in the denominator.Marking to market reprices the bond every day based on the yield curve: at maturity, it is worth the contracted rate; before then, it is worth the day's market price.Longer maturities amplify price sensitivity — the intuition that will become duration in Lesson 7.In the next lesson, we move beyond pure fixed-rate bonds: we will price the LFT (Tesouro Selic) and Tesouro IPCA+ (NTN-B) — and understand why one of them is barely affected by marking to market while the other is the most sensitive of all.Exercise for classroom discussionThe case of the panicked investor. Let us think it through together.In March, Marina bought R$ 20,000 in Tesouro Prefixado 2031 at a rate of 12.5% a year, planning to redeem it at maturity to make a down payment on a property. In August, after a deterioration in the fiscal outlook, the rate for that maturity rose to 14.8%. Marina opens the app, sees her position valued at R$ 18,400 and sends a message to the family group: "I lost R$ 1,600 on the safest investment in Brazil! I'm going to sell everything before things get worse!"Discuss with your group:(a) Did Marina lose R$ 1,600? In what sense did she, and in what sense did she not?(b) What happens if she sells today? And if she holds it until 2031?(c) Does "selling before things get worse" make sense for someone who plans to hold the bond until maturity? What would have to happen for selling early to be the right decision?(d) If, instead of rising, the rate had fallen to 10%, what would the app show? Should Marina celebrate and sell?(e) What question should Marina have asked herself BEFORE choosing a long-term fixed-rate bond for a goal with a set date?Guidance for the teacher: (a) The loss is real as an exit price today (marking to market), but it is only potential for someone who holds the bond: at maturity, the R$ 1,000 per bond has not changed — both meanings should be made explicit. (b) Selling crystallizes the loss at the market price; holding until 2031 produces exactly the contracted 12.5% per year. (c) No — selling "before things get worse" is justified only if she needs the money sooner, or if she believes she can reallocate it at a rate that offsets the crystallized loss; for a goal at maturity, the fluctuations are noise. (d) It would show a profit above the contracted return; selling and realizing the gain may make sense, but reinvesting would mean doing so at lower rates (10%) — there is no free lunch. (e) Matching the bond's maturity to the goal's horizon, plus the ability to tolerate seeing the statement fluctuate. Conclusion: the villain in the story is not the bond, but the mismatch between the product and the investment horizon — and marking to market is transparency, not a flaw.Exercises1) (Original - CESGRANRIO style) An LTN maturing in 252 business days is trading at a rate of 12% a year. The approximate unit price of this bond is:A) R$ 892.86.B) R$ 880.00.C) R$ 907,03.D) R$ 797,19.E) R$ 1.120,00.2) (Original - CESGRANRIO style) An LTN maturing in 126 business days (half a year in business days) is traded at 12% per year. Using the convention of 252 business days and compound capitalization, its approximate unit price is:A) R$ 944,91.B) R$ 940,00.C) R$ 892,86.D) R$ 887,79.E) R$ 943,40.3) (Original - CESGRANRIO style) An investor paid R$ 850,00 for an LTN maturing in exactly 252 business days. The annual rate implicit in the transaction is approximately:A) 17,6% per year.B) 15,0% per year.C) 8,5% per year.D) 12,0% per year.E) 21,3% per year.4) (Original - FGV style) An investor purchased a 2-year LTN at a rate of 12% per year. One week later, the market rate for the same maturity fell to 10% per year. Regarding the investor's position, it is correct to state that:A) the market price of the bond rose, generating a potential gain from mark-to-market accounting.B) the market price of the bond fell, generating a potential loss from mark-to-market accounting.C) the market price remained unchanged because the purchase rate is guaranteed.D) the yield at maturity became 10% per year.E) the bond was automatically redeemed by the National Treasury.5) (Original - CESGRANRIO style) Regarding the mark-to-market valuation of fixed-rate government bonds, it is correct to state that:A) an investor who holds the bond until maturity receives the rate agreed at purchase, regardless of intermediate price fluctuations.B) the agreed return is guaranteed to the investor on any sale date.C) mark-to-market valuation applies only to variable-income securities.D) the LTN's face value at redemption is adjusted daily according to the yield curve.E) declines in market interest rates reduce the price of fixed-rate bonds held in the portfolio.6) (Original - CESGRANRIO style) Two LTNs, maturing in 1 year and 5 years, are subject to the same 2-percentage-point increase in market rates. Comparing the effects on their prices:A) the 5-year LTN will suffer a greater percentage decline because longer maturities increase the price's sensitivity to interest rates.B) the 1-year LTN will suffer a greater percentage decline because it matures sooner.C) both will suffer the same percentage decline because the rate shock is identical.D) neither will decline because the nominal value of R$ 1.000 is fixed.E) the 5-year LTN will appreciate, offsetting the rate increase.7) (Original - CEBRASPE style, judge the statement) Regarding fixed-rate government bonds, judge the following statement: "Because its return is defined at the time of purchase, an LTN does not expose the investor to the risk of a nominal loss if the investor sells the bond before maturity."( ) True ( ) False8) (Original - CEBRASPE style, judge the statement) Judge the following statement: "When pricing an LTN, the nominal value of R$ 1.000,00 is discounted at the negotiated rate, taking into account the number of business days until maturity on an annual basis of 252 business days."( ) True ( ) FalseAnswer Key1) A — P = 1.000/(1,12)^(252/252) = 1.000/1,12 = 892,86. Option B (R$ 880,00) reflects a simple-interest discount — a classic mistake; option D is the price for 2 years, not 1.2) A — P = 1.000/(1,12)^(126/252) = 1.000/(1,12)^0,5 = 1.000/1,0583 = 944,91. Half a period under the compound convention uses an exponent of 0,5, not half the rate (which would yield option E, 1.000/1,06 = 943,40... notice how close the values are: the exam board is testing precisely this proximity).3) A — i = (1.000/850) − 1 = 0,1765 → 17,6% per year. The greater the discount, the higher the implicit rate.4) A — The rate fell from 12% to 10%, so the price rose (from R$ 797,19 to R$ 826,45), generating a potential gain. For investors who hold the bond until maturity, however, the return remains the agreed 12% — option D confuses the market rate with the purchase rate.5) A — This is the golden rule for fixed-rate investments. Option D is a major trap: mark-to-market valuation adjusts the bond's PRICE daily; the nominal redemption value (R$ 1.000) never changes.6) A — Same shock, larger exponent: compound discounting magnifies the effect of the rate over longer maturities. This is the intuition behind duration (Lesson 7).7) False — Before maturity, the LTN is worth its market price on that day; if rates have risen, an early sale can result in a loss, including a nominal loss. The agreed return is guaranteed only at maturity.8) True — This is exactly the formula P = 1.000/(1+i)^(du/252), using the Brazilian convention of 252 business days and compound capitalization.
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